Assignment 4 :: Concurrencies of a Triangle

The Nine Point Circle

By Jamie K. York

The nine-point circle is a unique relation between the altitudes of the triangle, the midpoints of its sides, and the midpoints between the orthocenter and each vertex.

Begin by finding the altitudes of the given triangle, marking the intersection of each altitude with the side of the triangle. Note that these three lines intersect inside the triangle at a single point called the orthocenter.

Now, find the midpoints on each side of the triangle.

Also find the midpoints between the orthocenter and each vertex.

These nine points are those that will align on the circumference of the constructed circle. In order to construct the circle properly, we will first find the center. To find the center, create an inscribed triangle by connecting the midpoints of the sides of the original triangle. Find the center of the inscribed triangle by connecting the perpendicular bisectors. The center of the inscribed triangle is also the center of the nine-point circle. We can now more accurately construct the circle.

Further Exploration

The triangle used above is acute. Can the nine-point circle be constructed the same way with a right or obtuse triangle?

Check out this animation to see more: GSP Nine-Point Circle Animation